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Dyson Brownian motion model
As we have seen in section (I.6), thanks to the invariance under rotations, RMT provides an explicit expression for the joint probability distribution P(x1, . . . , xN) of the eigenvalues of N × N matrices belonging to the three classical Gaussian ensembles [106]: P(x1, . . . , xN) = CN e−W, (II.4.1) W = 12 XN j=1 x2j − X i<j log |xi − xj | , (II.4.2) where = 1, 2 or 4 respectively for the GOE, GUE and GSE case (see section (I.6)). This expression is identical to the probability density of the position of N unit charges on an infinite straight line −1 < x < 1 subjected to the potential energy (I.6.2). If the parameter is identified with the inverse temperature = (KBT)−1, the computation of averages over the distribution (II.4.1) is equivalent to the computation of thermodynamic quantities. With this picture, referred to as the Coulomb gas model, the problem of averaging over Gaussian ensembles is mapped into a statistical mechanics model. Dyson extended this idea, in such a way that the Coulomb gas model acquires meaning not only on a thermodynamic point of view, but also as a dynamical system out of equilibrium: to do that, the variables xj must be interpreted as positions of particles in Brownian motion [209, 210, 211]. Each particle has therefore no inertia and is subjected to a fluctuating force fj and to a frictional force proportional to the velocity: the motion of the particles is thus described by the following system of Langevin equations.
The recursion equation for the resolvent
In section (II.2) we have presented the cavity approximation, which is based on the assumption that the Gaussian probability measure (II.2.2), used to write the resolvent in the form of a Gaussian integral over auxiliary fields i, factorizes over the sites of cavity graph: this assumption is exact on the Bethe Lattice (see section (II.2.1)), thanks to the particular tree-like structure in which loops are absent. We can show that such assumption is justified also in the case of Lévy Matrices, using the fully-connected structure of the model and the generalized central limit theorem.
The starting point is the expression of the resolvent (II.2.1) of a system of size N in terms of a Gaussian integral over auxiliary fields i. Following the authors of Ref. [38] we can imagine to add a row k and its symmetric columns to the matrix H: this, in terms of the tight binding representation of the matrix is equivalent to add a site to the system. For the resolvent of the system with (N + 1) sites, if we integrate over all fields except k, we obtain.
Computation of the mobility edge
As explained above, one of the most interesting features of Lévy matrices is the presence of a mobility edge in the spectrum, separating the extended states from the localized ones. We present here a procedure to obtain the transition point in the spectrum, based on studying the stability of the localized phase under a perturbation obtained adding a small imaginary part to the resolvent. This is the approach used by Abou Chacra, Anderson and Thouless to study the localization transition for the Anderson Model on the Bethe Lattice [8]. The criterion is based on the different behavior of the distribution of the imaginary part of the diagonal elements of the resolvent G(E) in the two phases: if the energy E corresponds to a delocalized states, =G(E − i) has a finite typical value when ! 0, which is of order one. On the contrary, if E belongs to the portion of the spectrum which is localized, the typical value of =G(E) tends to zero when ! 0. Since in the localized phase there is a discrete set of sites where =G(E) is infinitely large, therefore the mean value of =G(E) is however finite. The stability of the localized phase is governed by an eigenvalue equation for the same integral operator found in Ref. [38]: here we show how the analysis of such equation can be simplified, and we identify the mobility edge. We start from the equations (III.3.1) for the real and imaginary part of the self-energy, which we can rewrite in the form.
The mapping to directed polymers in random media
The result we have found for the mobility edge can be obtained, always starting from the recursion equation (III.4.2), in another way, making an analogy with the problem of directed polymers in random media [237, 238]. In particular, we can notice that the equation (III.4.2) for the imaginary part of the self energy can be rearranged in a form which coincides with the expression of the partition function of directed polymers in random media. We proceed replacing the js appearing in the sum of the right hand side of the equation (III.4.2) by their expression in terms of their “neighbors”: if we do that one time we obtain the expression i1 = X i26=i1 xi1i2 (E − Si2)2 i2 .
Table of contents :
Introduction
I The Anderson Localization Transition: introduction
I.1 Disorder and Localization
I.2 Characteristics of the transition and localized states
I.3 Anderson Localization in one and two dimensions and weak localization
I.4 Scaling theory and field theory formulation
I.5 Anderson transition on the Bethe Lattice: mean field
I.6 Anderson Localization and Random Matrix Theory
I.7 Brief review on numerical results
I.8 Experiments on Localization
I.9 Open problems
I.9.1 Interactions and Many Body Localization
I.9.2 The problem of the intermediate phase
II Overview on analytical techniques and known results
II.1 Definitions
II.2 Cavity equations
II.2.1 On the Bethe Lattice
II.3 Supersymmetric method
II.4 Dyson Brownian motion model
IIILocalization Transitions of Lévy Matrices
III.1 Introduction and motivations
III.2 The recursion equation for the resolvent
III.3 The density of states
III.4 Computation of the mobility edge
III.4.1 The mapping to directed polymers in random media
III.5 Numerical check of the phase diagram
III.6 The problem of the intermediate phase: previous results
III.7 The Supersymmetric method applied to Lévy Matrices
III.7.1 Equation on R() with the supersymmetric method
III.8 The Dyson Brownian motion argument
III.9 Numerical results for μ 2 (1, 2)
III.10 Numerical results for μ 2 (0, 1)
III.10.1 Numerical results for Q(G)
III.10.2 Wavefunction statistics and multifractal spectrum
III.11 Summary of the results
IV Critical properties of the Anderson model in high dimension
IV.1 Numerical results in d = 3, . . . , 6
IV.1.1 Transport properties
IV.1.2 Statistics of level spacings and of wave-functions coefficients
IV.2 Strong Disorder RG
IV.3 Summary of the results
Conclusion and perspectives
A Transfer Matrix, conductance and localization length
A.1 Transfer Matrix and conductance
A.2 Transfer Matrix and localization length
B Critical wave functions: multifractality
C The IPR in terms of the Green function
D The DoS of Sparse RM model with the supersymmetric method
E Meaning of the order parameter function
F The generalized central limit theorem
G Computation of the mobility edge of Lévy Matrices
List of Figures
Bibliography
